American Institute of Mathematical Sciences

Advanced
2015, 2015(special): 1034-1040. doi: 10.3934/proc.2015.1034

Recurrence of multi-dimensional diffusion processes in Brownian environments

Hiroshi Takahashi 1, and  Yozo Tamura 2,

1. 

Colledge of Science and Technology, Nihon University, Narashino-dai, Funabashi 274-8501, Japan
2. 

Faculty of Science and Technology, Keio University, Hiyoshi, Yokohama 223-8522, Japan

Received  September 2014 Revised  February 2015 Published  November 2015

We consider limiting behavior of multi-dimensional diffusion processes in two types of Brownian environments. One is given values at different $d$ points of a one-dimensional Brownian motion, which is supposed to be a multi-parameter environment, and other is given by $d$ independent one-dimensional Brownian motions. We show recurrence of multi-dimensional diffusion processes in both Brownian environments above for any dimension and almost all environments. Their limiting behavior is quite different from that of ordinary multi-dimensional Brownian motion. We also consider cases of reflected Brownian environments.
Keywords: recurrence, transience., Diffusion process in Brownian environment.
Mathematics Subject Classification: Primary: 60K37, 60J60; Secondary: 60J6.
Citation: Hiroshi Takahashi, Yozo Tamura. Recurrence of multi-dimensional diffusion processes in Brownian environments. Conference Publications, 2015, 2015 (special) : 1034-1040. doi: 10.3934/proc.2015.1034
References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium. Ann. Probab., 14 (1986), 1206-1218.  Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization, in "Stochastic Processes - Mathematics and Physics II (Bielefeld, 1985)'' (eds. S. Albeverio, Ph. Blanchard and L. Streit), Lect. Notes in Math., 1250, Springer-Verlag, (1987), 87-97.  Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes. Publ. RIMS, Kyoto Univ., 14 (1978), 441-486.  Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process. Proc. Imp. Acad., 20 (1944), 54-55.  Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'' Springer-Verlag, New York, 1965.  Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments, J. Korean Math. Soc., 48 (2011), 147-158.  Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45-106.  Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment, Probab. Theory Related Fields, 99 (1994), 549-580.  Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment, Theory Probab. Appl., 27 (1982), 256-268.  Google Scholar

[10]

F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31.  Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments. Statist. Proba. Lett., 69 (2004), 171-174. Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments, in "Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986)'' (ed. G. Papanicolau), IMA Vol. Math. Appl. 9. Springer, (1987), 189-210.  Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377-381.  Google Scholar

show all references

References:
[1]

T. Brox, A one-dimensional diffusion process in a Wiener medium. Ann. Probab., 14 (1986), 1206-1218.  Google Scholar

[2]

M. Fukushima, S. Nakao and M. Takeda, On Dirichlet form with random date - recurrence and homogenization, in "Stochastic Processes - Mathematics and Physics II (Bielefeld, 1985)'' (eds. S. Albeverio, Ph. Blanchard and L. Streit), Lect. Notes in Math., 1250, Springer-Verlag, (1987), 87-97.  Google Scholar

[3]

K. Ichihara, Some global properties of symmetric diffusion processes. Publ. RIMS, Kyoto Univ., 14 (1978), 441-486.  Google Scholar

[4]

K. Itô, On the ergodicity of a certain stationary process. Proc. Imp. Acad., 20 (1944), 54-55.  Google Scholar

[5]

K. Itô and H.P. McKean,Jr., "Diffusion Processes and Their Sample Paths,'' Springer-Verlag, New York, 1965.  Google Scholar

[6]

D. Kim, Some limit theorems related to multi-dimensional diffusions in random environments, J. Korean Math. Soc., 48 (2011), 147-158.  Google Scholar

[7]

G. Maruyama, The harmonic analysis of stationary stochastic processes, Mem. Fac. Sci. Kyushu Univ., A4 (1949), 45-106.  Google Scholar

[8]

P. Mathieu, Zero white noise limit through Dirichlet forms, with application to diffusions in a random environment, Probab. Theory Related Fields, 99 (1994), 549-580.  Google Scholar

[9]

Y. Sinai, The limit behavior of a one-dimensional random walk in a random environment, Theory Probab. Appl., 27 (1982), 256-268.  Google Scholar

[10]

F. Solomon, Random walks in a random environment, Ann. Probab., 3 (1975), 1-31.  Google Scholar

[11]

H. Takahashi, Recurrence and transience of multi-dimensional diffusion processes in reflected Brownian environments. Statist. Proba. Lett., 69 (2004), 171-174. Google Scholar

[12]

H. Tanaka, Limit distributions for one-dimensional diffusion process in self-similar random environments, in "Hydrodynamic Behavior and Interacting Particle Systems (Minneapolis, Minn., 1986)'' (ed. G. Papanicolau), IMA Vol. Math. Appl. 9. Springer, (1987), 189-210.  Google Scholar

[13]

H. Tanaka, Recurrence of a diffusion process in a multi-dimensional Brownian environment, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 377-381.  Google Scholar

[1]

Yan Wang, Lei Wang, Yanxiang Zhao, Aimin Song, Yanping Ma. A stochastic model for microbial fermentation process under Gaussian white noise environment. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 381-392. doi: 10.3934/naco.2015.5.381
[2]

Dang H. Nguyen, George Yin. Recurrence for switching diffusion with past dependent switching and countable state space. Mathematical Control & Related Fields, 2018, 8 (3&4) : 879-897. doi: 10.3934/mcrf.2018039
[3]

Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control & Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[4]

Harish Garg. Novel correlation coefficients under the intuitionistic multiplicative environment and their applications to decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1501-1519. doi: 10.3934/jimo.2018018
[5]

Genni Fragnelli, A. Idrissi, L. Maniar. The asymptotic behavior of a population equation with diffusion and delayed birth process. Discrete & Continuous Dynamical Systems - B, 2007, 7 (4) : 735-754. doi: 10.3934/dcdsb.2007.7.735
[6]

Arnaud Debussche, Sylvain De Moor, Julien Vovelle. Diffusion limit for the radiative transfer equation perturbed by a Wiener process. Kinetic & Related Models, 2015, 8 (3) : 467-492. doi: 10.3934/krm.2015.8.467
[7]

Shin-Ichiro Ei, Hiroshi Matsuzawa. The motion of a transition layer for a bistable reaction diffusion equation with heterogeneous environment. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 901-921. doi: 10.3934/dcds.2010.26.901
[8]

Henri Berestycki, Guillemette Chapuisat. Traveling fronts guided by the environment for reaction-diffusion equations. Networks & Heterogeneous Media, 2013, 8 (1) : 79-114. doi: 10.3934/nhm.2013.8.79
[9]

Harish Garg. Some robust improved geometric aggregation operators under interval-valued intuitionistic fuzzy environment for multi-criteria decision-making process. Journal of Industrial & Management Optimization, 2018, 14 (1) : 283-308. doi: 10.3934/jimo.2017047
[10]

Chihurn Kim, Dong Han Kim. On the law of logarithm of the recurrence time. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 581-587. doi: 10.3934/dcds.2004.10.581
[11]

Petr Kůrka, Vincent Penné, Sandro Vaienti. Dynamically defined recurrence dimension. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 137-146. doi: 10.3934/dcds.2002.8.137
[12]

Serge Troubetzkoy. Recurrence in generic staircases. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 1047-1053. doi: 10.3934/dcds.2012.32.1047
[13]

Michael Blank. Recurrence for measurable semigroup actions. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1649-1665. doi: 10.3934/dcds.2020335
[14]

Michel Benaim, Morris W. Hirsch. Chain recurrence in surface flows. Discrete & Continuous Dynamical Systems, 1995, 1 (1) : 1-16. doi: 10.3934/dcds.1995.1.1
[15]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[16]

Milton Ko. Rényi entropy and recurrence. Discrete & Continuous Dynamical Systems, 2013, 33 (6) : 2403-2421. doi: 10.3934/dcds.2013.33.2403
[17]

Miguel Abadi, Sandro Vaienti. Large deviations for short recurrence. Discrete & Continuous Dynamical Systems, 2008, 21 (3) : 729-747. doi: 10.3934/dcds.2008.21.729
[18]

Isabelle Kuhwald, Ilya Pavlyukevich. Bistable behaviour of a jump-diffusion driven by a periodic stable-like additive process. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3175-3190. doi: 10.3934/dcdsb.2016092
[19]

Wuyuan Jiang. The maximum surplus before ruin in a jump-diffusion insurance risk process with dependence. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3037-3050. doi: 10.3934/dcdsb.2018298
[20]

Yejuan Wang, Peter E. Kloeden. The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain. Discrete & Continuous Dynamical Systems, 2014, 34 (10) : 4343-4370. doi: 10.3934/dcds.2014.34.4343

 Impact Factor: 

Tools

Metrics

  • PDF downloads (149)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy